The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces

Abstract

This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0, 1] of functions from [0, 1] to the reals with the supremum metric is given. An answer is given to a question of Sierpinski on isometric embeddings of U in C[0, 1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry. The question of what Banach spaces can be embedded in a linear isometric fashion in this uniquely determined closed linear span of U is investigated. 0. Introduction. A well-known result of elementary topology is the fact that certain topological spaces, such as the Hilbert cube, are “universal separable metric spaces”—such a space is a separable metric space which contains a homeomorph of each separable metric space. A less well-known result of metric topology is that there are universal separable metric spaces up to isometry ; that is, there are separable metric spaces which contain an isometric copy of each separable metric space. The best-known theorem establishing the existence of a universal separable metric space up to isometry is the theorem of Banach and Mazur which asserts that C[0, 1], the space of continuous functions from [0, 1] to the reals with the supremum metric, is such a space (see [B], [B-P]; Banach and Mazur actually show that C[0, 1] is a universal separable Banach space up to linear isometry; we cite this result as Theorem 4 of Part III). But the first result of this kind was obtained by Urysohn (see [U]), who constructed a metric space U of this kind which he proved can be characterized up to isometry by the fact that it is a universal separable metric space up to isometry which is complete and “metrically homogeneous with respect to finite sets”. The definition of this property is given below. Urysohn’s construction of U is highly abstract. We are able to present a more concrete construction of U inside C[0, 1] (in the proof of Theorem 1 of